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Fearless Symmetry by Robert Gross, Avner Ash

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72 CHAPTER 7
In other words,
a
p
=−1ifS(F
p
) contains no elements.
a
p
= 0ifS(F
p
) contains one element. This happens only
if a is a multiple of p.
a
p
= 1ifS(F
p
) contains two elements.
Because
a
p
depends only on a modulo p,
a
p
=
a + kp
p
for any integer k.
4
You might wonder why Legendre chose to shift by 1, rather than
just letting
a
p
be the number of elements in S(F
p
). One answer is
the following easy pair of equations:
0
p
= 0,
1
p
= 1.
But there is more to it than that.
Digression: Notation Guides Thinking
There are some very famous instances of how proper choices of
notation can help speed the progress of mathematics. Someone
outside of the field might not even realize that there is a distinction
between the notations that mathematicians use and the subject
itself.
For example, algebraic notation—x, y, and all the rest—is
notation, not mathematics. Greek mathematicians did not have
that symbolism, and they were able to express problems in words
and solve them without using symbols. Today, of course, the use of
x to stand for an unknown thing is so widespread that it has passed
into popular culture. Using letters to stand for quantities makes it
4
Pay careful attention to this fact; we will be using it repeatedly, both in proofs and
computations.

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